The present invention relates to methods and instruments for fluorescence detection for use in electrophoretic instruments and electrophoresis that are applied to analyzing biopolymers such as nucleotides and proteins.
Electrophoresis in which to detect fluorescence by laser induction is widely used as one fundamental technique for analyzing biopolymers such as nucleotides and proteins because of its high sensitivity and convenience. In the biopolymer analysis field, capillary electrophoresis has lately been used commonly, superceding slab gel electrophoresis that was a mainstream analysis method. For the capillary electrophoresis, less Joule heating is generated when analytes electophoretically migrate and therefore high voltage can be used. As a result, analysis can be performed in a shorter time. The length of a migration (separation) channel is generally 50 cm to 20 cm. Aiming at reducing analysis time and downsizing the analysis system, diverse techniques have been developed to shorten the migration (separation) channel.
Such a method is described in Science, 261, 895-897 (1993) (prior art-1) that, by forming capillary channels on a substrate by application of photolithography technique and making analytes electrophoretically migrate through the channels, separating a plurality of fluorescence labeled amine acid is achieved with migration (separation) channel length of 0.75 cm to 2.2 cm.
Under conditions that unimolecular detection is performed, several species of deoxyribonucleic acid (DNA) can be separated in a migration (separation) channel with effective length of 0.25 mm, which is described in Anal. Chem., 67, 3181-3186 (1995) (prior art-2).
A method in which fluorescence emitted from analytes migrating through channels is sequentially detected through 55 300 xcexcm-wide detection slits arranged at intervals of 700 xcexcm and the velocity of the analytes is measured by Fourier transform of fluorescence intensity is described in Anal. Chem., 71, 2130-2138 (1999) (prior art-3).
A method in which interference fringes of excitation light are generated across analytes including fluorescent material and fluctuation of fluorescence radiated from the fluorescent material is measured in a correlation function, thereby measuring the fluid velocity of the analytes (one type of the method called Fluorescence Correlations (FCS)) is described in Kokai (Japanese Unexamined Patent Publication) No. Sho-53-40586 (No. 40586 of 1978) (prior art-4).
Using the analysis apparatus configured in the same way as for prior art 1, when the migration channel length is made shorter by 5 mm, the number of theoretical plates will be 5800, which is described in Anal. Chem., 65, 2637-2642 (1993) (prior art-5).
Under general conditions, electrophoresis of prior art posed the following problem. Band broadening of analytes injected into the sample inlet end of a channel in which electrophoresis takes place restricts separation of the analytes. When analytes with close mobilities are used, it is difficult to achieve good separation in a migration (separation) channel with length of the order of millimeters. Actually, in prior art-1, a 2.2-cm long migration (separation) channel separates analytes with mobility difference of 10% or less, whereas a 0.75-cm long channel can separate only analytes with mobility difference of 20% or more.
Prior art-3 also presents problems. Detection slit pitch is restricted by the band broadening of injected analytes and cannot be reduced unlimitedly. The number of detection slits cannot be reduced arbitrarily because it influences separation performance. In prior art-3, the pitch (clearance) between detection slits is 0.7 mm, the number of detection slits is 55, and the effective migration (separation) channel length is about 4 cm which is longer than the channel length in prior art-1.
In prior art-1 and -3, because analytes in narrow band broadening are injected, it is necessary to form two or more crossing channels on the substrate.
Application of the FCS technique described in prior art-4 to the electrophoresis field has not been reviewed heretofore. The present invention is made through consideration of improving the FCS technique described in prior art-4 and applying it to electrophoresis.
The object of the present invention is to provide methods and instruments for florescence detection, enabling better separation and detection of a plurality of species of analytes with different mobility in a migration (separation) channel with length of the order of millimeters without being restricted by band broadening of the analytes injected into the sample injection end of the channel, thereby solving the above-described problems.
Methodology for fluorescence detection of the present invention is as follows. Analytes are caused to electophoretically migrate in a migration channel such that the analytes disperse in succession across a detection region where they are detected. Excitation light is applied to the detection region. The excitation light is controlled to have an intensity profile that periodically changes in a cycle equaling a pitch greater than the size of a analyte molecule in the direction that the analytes move (in the direction of electric field application). Instead, a slit is located between the detection region and a detector for detecting fluorescence. The slit is designed to have a transmittance profile that periodically changes in a cycle equaling a pitch greater than the size of a analyte molecule. For detected fluorescence emission from the analytes in the detection region, its power spectrum is obtained. Alternatively, an array sensor is used as the detector for detecting fluorescence and fluorescence emission from the detection region is measured. Distribution in the migration direction appears in the fluorescence measurements. Calculation is executed for the sum of the products of fluorescence intensity detected by the photoelectric elements of the array sensor and a function of predetermined pitch and the power spectrum as the sum of the products is obtained.
Description of the Principle Underlying the Invention
First, the principle underlying the invention is now described hereinafter. On the assumption that fluorescence labeled analytes of one species are irradiated by monochromatic excitation light, we consider that fluorescence emitted from the analytes exposed to the excitation light is detected by a photomultiplier (PM). Output current i (t) of the PM is expressed by mathematical expression 1 using molarity C(r, t) of analyte at time t and position r=(x, y, z), and the product I(copyright) of multiplying the following excitation light intensity at position r, and efficiency of collection of fluorescent light emitted from an analyte on the photoelectric plane of the PM.
Mathematical Expression 1
i(t)={gexcex7xcex5 Qln10/(hc/xcex)}∫I(r)C(r, t)drxe2x80x83xe2x80x83[Mathematical expression 1]
where dr=dxdydz, xcex5 is a molar excitation coefficient of analyte, Q is a fluorescence quantum yield of analyte, h is a plank constant, c is the velocity of light, xcex is wavelength of the excitation light, xcex7 is quantum efficiency of the PM, e is elementary electric charge, g is current gain of the PM. By setting g=1, mathematical expression 1 can be applied to photodiodes. FCS (Fluorescence Correlations) is based on analysis of fluctuation xcex4i (t)=i (t)xe2x88x92 less than i (t) greater than  of i (t) when the concentration distribution of the analytes stays in a thermal equilibrium state. In this relation,  less than X (t) greater than  is an average in the ensemble of X (t). A normalized auto-correlation function G (t) that represents fundamental quantity of time dependency of fluctuation xcex4i (t) is defined by mathematical expression 2.
Mathematical Expression 2
G(t)= less than xcex4i(0)xcex4i(t) greater than / less than (xcex4i(t))2 greater than xe2x80x83xe2x80x83[Mathematical expression 2]
Furthermore, to analyze fluctuation xcex4i (t) in the frequency domain, a normalized power spectrum S (xcexd) of xcex4i (t) is defined as power spectrum xcex4i (t)/ less than (xcex4i (t)2 greater than 1/2 (to be more precise, one-sided power spectrum). According to a Wiener-Khintchine theorem, S (xcexd) is expressed by mathematical expression 3. For integration ∫, the lower limit is 0 and the upper limit is
Mathematical Expression 3
S(xcexd)=4∫G(t) cos (2xcfx80xcexdt)dtxe2x80x83xe2x80x83[Mathematical expression 3]
In the following, we will consider dispersion with a Gaussian envelope having width L (exe2x88x922 wide) in the x-axis direction, sinusoidal oscillation by pitch p, and Gaussian profiles having width W (exe2x88x922 wide) and H (exe2x88x922 wide) in the y and z directions, respectively, as I (r) that is expressed by mathematical expression 4, provided constraint that is specified in mathematical expression 5 shall be fulfilled.
Mathematical Expression 4
I(r)=I0exp{xe2x88x928(x/L)2}{cos (2xcfx80x/p)+1}xc3x97exp{xe2x88x928(y/W)2}exp{xe2x88x928(z/H)2}xe2x80x83xe2x80x83[Mathematical expression 4]
Mathematical Expression 5
1 xcexcm less than p less than  less than L, W, Hxe2x80x83xe2x80x83[Mathematical expression 5]
For example, dispersion represented by I (r) can be realized in this way. Create an interference pattern (fringes) by making two elliptic Gaussian beams that are symmetrical with regard to the y axis of the x-y plane on which the optical axis is placed intersect each other in a sample cell located in the vicinity of the origin. Detect fluorescence through slits having a Gaussian transmittance profile from the z-axis direction toward the y direction. If analytes move in the x-axis direction at a constant velocity V and constraint that is specified in mathematical expression 6 for a translational diffusion coefficient D of the analytes is fulfilled, G (t) to be obtained from I (r) in mathematical expression 4 is given by mathematical expression 7. Derivation of mathematical expression 7 will be fully described later.
Mathematical Expression 6
D less than  less than xcexd/4xe2x80x83xe2x80x83[Mathematical expression 6]
Mathematical Expression 7
G(t)=(⅓){cos (2xcfx80Vt/p)+1}xc3x97exp{xe2x88x92(2Vt/L)2xe2x88x92D(2xcfx80/p)2t}+(⅔)exp{xe2x88x92(2Vt/L)2}xe2x80x83xe2x80x83[Mathematical expression 7]
As implied by the first term of the right member of mathematical expression 7, G (t) includes a component oscillating at frequency V/p in proportion to the velocity V of analyte. Thus, power spectrum S (xcexd) has a frequency peak proportional to the velocity of analyte. If the velocity V of analyte is determined, resulting from electrophoresis, the graph of S (xcexd) corresponds to a conventional electropherogram.
FIG. 1 is a schematic diagram for explaining the principle underlying the present invention. Analyte samples 1xe2x80x941 to 1-N disperse across the whole detection region of a migration channel 2 filled with a sieving matrix (buffer solution, gel, polymers, etc.) Electrodes 4 and 5 located in contact with a power supply 3 and the sieving matrix cause the analytes to electophoretically migrate at predetermined velocity V. The detection region is irradiated by excitation light 6 whose intensity changes in a cycle equaling pitch p in the direction that the analytes move. Fluorescence emission from the analytes exposed to the excitation light is detected by a detector 7. Fluctuation xcex4i (t) of output current from the detector 7 is analyzed by a spectrum analyzer 8 and the obtained spectrum is displayed.
A typical method for approximately realizing periodically changing I (r) as represented in mathematical expression 4 is roughly divided into the following three ways.
(1) Interference fringes are generated to cause periodical change in the intensity profile of excitation light as shown in FIG. 1.
(2) A slit with its transmittance changing periodically is installed between the detection region and the detector and fluorescence emission from the analytes is detected through the slit.
(3) An image sensor is used as the detector for detecting fluorescence and fluorescence intensity distributed in time and space is measured and profiled separately. Determine C (r, t) and calculate the right member of mathematical expression 1 for I (r) that changes periodically.
Description of Separation Efficiency in the Present Invention
If a plurality of species of analytes that move at different velocity is employed, the PM output current and its fluctuation will be the sum of photo current produced by each analyte and its fluctuation. If the analytes are sufficiently dilute and their mutual action is negligible, the power spectrum of the sum of fluctuation will be the sum of the power spectrum of fluctuation specific to each analyte. Thus, under proper conditions, peaks with different center frequency as many as the number of the analyte species can be separated and detected on the graph of S (xcexd). In the following, efficiency of analyte separation according to the methodology of the present invention will be discussed. In view hereof, the number of theoretical plates (NTP) is employed as an index of separation efficiency. NTP is defined by mathematical expression 8 using full width at half maximum (FWHM) of a peak that is positioned at V/p of S (xcexd).
Mathematical Expression 8
NTP=(81n2){V/(pxc3x97FWHM)}2xe2x80x83xe2x80x83[Mathematical expression 8]
As evident from mathematical expression 7, because S (xcexd) is Gaussian and Lorentzian convolution, it is difficult to represent FWHM analytically. Thus, first define coherence time xcfx84c as time at which the amplitude of the oscillating part in the first term of the right member of mathematical expression 7 becomes equaling exe2x88x921. Then, approximate S (xcexd) by means of pure Gaussian Fourier transform having the same coherence time. According to this approximation, NTP is obtained by mathematical expression 9 where p0 is defined by mathematical expression 10.
Mathematical Expression 9
(NTP)1/2=21/2xcfx80xcfx84cV/p=xcfx802xe2x88x921/2(p/p0)(L/p0)/[1+{1+(P/P0)4}1/2]xe2x80x83xe2x80x83[Mathematical expression 9]
Mathematical Expression 10
p0=xcfx80(DL/V)1/2xe2x80x83xe2x80x83[Mathematical expression 10]
A curve shown in FIG. 2(A) represents a function of (NTP)1/2/(L/p0) versus p/p0. From this curve, to achieve good separation under the conditions that the parameters other than p are fixed, pxe2x89xa0p0 is desirable and the maximum separation efficiency is obtained when (p/p0)=31/4.
For example, assuming L=1 mm and p=p0=10 xcexcm, NTPxe2x89xa0(L/p0)2/1.18 is obtained from mathematical expression 9; that is, NTPxe2x89xa08470 is attained, about 1.5 times the NTP of 5800 attained in a 5-mm migration channel in prior art-5.
For two peaks having equal FWHM, when the distance between the centers of the peaks is greater than FWHM/(21n2)1/2, the two peaks are generally defined as being completely separated in essence. When average mobility of two analytes is represented by xcexcAV and mobility difference between them xcex94xcexc, constraint that sets the conditions for achieving essentially complete separation of the two analytes having different mobility is expressed by mathematical expression 11. For example, if xcex94xcexc/xcexcAV=0.1, NTPxe2x89xa73600 must be fulfilled.
Mathematical Expression 11
NTPxe2x89xa736(xcexcAV/xcex94xcexc)2xe2x80x83xe2x80x83[Mathematical expression 11]
Description of Signal to Noise Ratio in the Present Invention
Next, signal to noise ratio (SNR) in the methodology of the present invention will be discussed. When an average number of molecules for the analytes detected is represented by N, we have mathematical expression 12.
Mathematical Expression 12
 less than {xcex4i(t)}2 greater than = less than {i(t)}2 greater than /Nxe2x80x83xe2x80x83[Mathematical expression 12]
The PM output current is input to a bandpass filter with pass-band width xcex94xcexd varying for a multiplicity of different center frequencies in a step for obtaining its power spectrum (at the present, actually, data sampling and discrete Fourier transform serve the function of such multi-filter). A root-mean-square of signal current is(t) output by a filter for center frequency xcexd is expressed by mathematical expression 13 where  less than  less than {X(t)}2 greater than  greater than  represents the root-mean-square of X (t).
Mathematical Expression 13
 less than  less than is(t)2 greater than  greater than = less than {xcex4i(t)}2 less than S(xcexd)xcex94xcexdxe2x80x83xe2x80x83[Mathematical expression 13]
For shot noise current iNS (t) and thermal noise current iNT (t), their root-mean-squares are expressed by mathematical expressions 14 and 15, respectively, where e is elementary electric charge, iB (t) is the sum of dark current of the PM and photo current induced by background emission other than fluorescence, kB is a Boltzman constant, T is absolute temperature, and R is load resistance connected to the filter output. According to mathematical expressions 13 to 15, SNR is obtained by mathematical expression 16.
Mathematical Expression 14
 less than  less than {iNS(t)}2 greater than  greater than =2ge{ less than i(t) greater than + less than iB(t) greater than }xcex94xcexdxe2x80x83xe2x80x83[Mathematical expression 14]
Mathematical Expression 15
 less than  less than {iNT(t)}2 greater than  greater than =4kBTRxe2x88x921xcex94xcexdxe2x80x83xe2x80x83[Mathematical expression 15]
Mathematical Expression 16
SNR= less than  less than {iS(t))}2 greater than  greater than /[ less than  less than {iNS(t)}2 greater than  greater than + less than  less than {iNT(t)}2 greater than  greater than ]=
xe2x80x83 less than {xcex4i(t)}2 greater than S(xcexd)/{2ge( less than i(t) greater than + less than iB(t) greater than )+4kBTRxe2x88x921}xe2x80x83xe2x80x83[Mathematical expression 16]
Assuming that  less than  less than {iNS(t)}2 greater than  greater than  greater than  greater than  less than  less than {iNT(t)}2 greater than  greater than  and  less than i(t) greater than  greater than  greater than  less than iB(t) greater than  and using relation specified in mathematical expression 12, we have mathematical expression 17 where iM is defined by mathematical expression 18. iM is photo current induced at the cathode of the PM by fluorescence of one analyte species.
Mathematical Expression 17
SNR=S(xcexd)iM/(2e)xe2x80x83xe2x80x83[Mathematical expression 17]
Mathematical Expression 18
iM= less than i(t) greater than /(gN)xe2x80x83xe2x80x83[Mathematical expression 18]
From mathematical expression 18, when N is extremely great, SNR is constant, not depending on N, and its value is determined by photo current per molecule. By approximation in the same manner as for evaluating separation efficiency, a maximum value of S (xcexd) where xcexd greater than 0, Smaxxe2x89xa0S (V/p) is expressed by mathematical expression 19.
Mathematical Expression 19
SMAX={xcfx801/2L/(6V)}xc3x97(p/p0)2/[1+{1+(p/p0)4}1/2]xe2x80x83xe2x80x83[Mathematical expression 19]
A curve shown in FIG. 2(B) represents dependency of Smax normalized with {(xcfx801/2L/(6V)} upon p/p0. In view of SNR, pxe2x89xa7p0 is desirable. Thus, we have mathematical expression 20 as the most desirable condition for making separation efficiency compatible with SNR.
Mathematical Expression 20
pxe2x89xa0p0xe2x80x83xe2x80x83[Mathematical Expression 20]
From FIGS. 2(A) and (B), to separate and detect different species of analytes without an extreme decrease of separation efficiency and SNR, constraint 1xe2x89xa6p/p0xe2x89xa65 must be fulfilled.
Detailed Description of Derivation of Mathematical Expression 7
In the following, derivation of mathematical expression 7 will be described in detail. Assuming that position vector r=r (x, y, z), dr=dxdydz, vector variable q=q (qx, qy, qz), dq=dqxdqydqz, and when spatial Fourier transform of I (r) is represented by I (q) that will be given in mathematical expression 21 and a correlation function of spatial Fourier transform of fluctuation in concentration xcex4 C (r, t) that will be given in mathematical expression 22 is represented by F (q, t) that will be given in mathematical expression 23, we have mathematical expression 24. In mathematical expressions 21 and 22, j is imaginary unit.
Mathematical Expression 21
I(q)=17I(r)exp(jqxc2x7r)drxe2x80x83xe2x80x83[Mathematical Expression 21]
Mathematical Expression 22
xcex4C(q, t)=∫xcex4C(r, t)exp(jqxc2x7r)drxe2x80x83xe2x80x83[Mathematical expression 22]
Mathematical Expression 23
F(q, t)= less than xcex4C(q, 0)xcex4C(q, t) greater than xe2x80x83xe2x80x83[Mathematical expression 23]
Mathematical Expression 24
G(t)=∫|I(q)|2F(q, t)dq/∫|I(q)|2F(q, 0)dqxe2x80x83xe2x80x83[Mathematical expression 24]
In the following, for calculation convenience, k=2xcfx80/p, "sgr"x=L/4, "sgr"y=W/4, "sgr"z=H/4, xcfx84i="sgr"i/D (i=x, y, z) are assigned. To redefine I (r) expressed in mathematical expression 7, by applying mathematical expressions 25 and 26, I (q) is expressed by mathematical expression 27.
Mathematical Expression 25
Ix(qx)=(xcfx80/2)1/2"sgr"x[exp{xe2x88x920.5"sgr"x2(qxxe2x88x92k)2}+exp{xe2x88x920.5"sgr"x2(qx+k)2}+, 2exp{xe2x88x920.5 ("sgr"xqx)2}xe2x80x83xe2x80x83[Mathematical expression 25]
Mathematical Expression 26
Ii(qi)=(2xcfx80)1/2"sgr"iexp{xe2x88x920.5("sgr"iqi)2}(i=y, z)xe2x80x83xe2x80x83[Mathematical expression 26]
Mathematical Expression 27
I(q)=I0Ix(qx)Iy(qy)Iz(qz)xe2x80x83xe2x80x83[Mathematical expression 27]
By definition as is given in mathematical expression 28 using imaginary unit j, F (q, t) is obtained as in mathematical expression 29.
Mathematical Expression 28
Fx(qx, t)=exp(jqxVtxe2x88x92qx2Dt), Fi(qi, t)=exp(xe2x88x92qi2Dt) (i=y, z)xe2x80x83xe2x80x83[Mathematical expression 28]
Mathematical Expression 29
F(q, t)=F(q, 0)Fx(qx, t)F(qy, t)F(qz, t)xe2x80x83xe2x80x83[Mathematical expression 29]
For q that is |q| greater than 106 from mathematical expression 5, I (q) will be substantially 0 by evaluating mathematical expressions 25 and 26. For q that is |q| greater than 106, F (q, 0) is regarded as constant, not depending on q, except for the vicinity of phase transition. Then, by defining Gi (t) in mathematical expression 30, G (t) is obtained as in mathematical expression 31.
Mathematical Expression 30
Gi(t)=∫|I(qi)|2Fi(qi, t)dqi/∫|I(qi)|2Fi(qi, 0)dqi(i=x, y, z)xe2x80x83xe2x80x83[Mathematical expression 30]
Mathematical Expression 31
G(t)=G(t)xG(t)yG(t)zxe2x80x83xe2x80x83[Mathematical expression 31]
From mathematical expression 25, |Ix (qx)| is obtained by mathematical expression 32, wherein because "sgr"xk greater than  greater than 1, the terms described within the brackets ([ ]) except the first three ones can be ignored. Then, by applying integration ∫ with the lower limit of xe2x88x92 and the upper limit of + and using imaginary unit j, B (k, t) is defined as in mathematical expression 33. From mathematical expressions 30 and 32, Gx (t) is obtained by mathematical expression 34.
Mathematical Expression 32
|Ix(qx)|=(xcfx80/2)"sgr"x2[exp{xe2x88x92"sgr"x2(qxxe2x88x92k)2}+exp{xe2x88x92"sgr"x2(qxxe2x88x92k)2}+exp{xe2x88x92"sgr"xxe2x88x92qx)2}+
2exp{xe2x88x92"sgr"x2(qxxe2x88x920.5k)2xe2x88x92("sgr"xk)2/4}+2exp{xe2x88x92"sgr"x2(qx+0.5k)2xe2x88x92("sgr"xk)2/4}]xe2x80x83xe2x80x83[Mathematical expression 32]
                                                                        B                ⁡                                  (                                      k                    ,                    t                                    )                                            =                            ⁢                              ∫                                                                            F                      x                                        ⁡                                          (                                              q                        ,                        t                                            )                                                        ⁢                  exp                  ⁢                                      {                                                                                            σ                          x                          2                                                ⁡                                                  (                                                      q                            -                            k                                                    )                                                                    2                                        }                                    ⁢                                      ⅆ                    q                                                                                                                          =                            ⁢                                                σ                  x                                      -                    1                                                  ⁢                                                      {                                          π                      /                                              (                                                  1                          +                                                      (                                                          t                              /                                                              τ                                x                                                                                      )                                                                          )                                                              }                                                        1                    /                    2                                                  xc3x97                                                                                                      ⁢                              exp                [                                                      {                                                                  j                        ⁢                                                  xe2x80x83                                                ⁢                        k                        ⁢                                                  xe2x80x83                                                ⁢                        V                        ⁢                                                  xe2x80x83                                                ⁢                        t                                            -                                                                        k                          2                                                ⁢                        D                        ⁢                                                  xe2x80x83                                                ⁢                        t                                            -                                                                        (                                                      V                            ⁢                                                          xe2x80x83                                                        ⁢                                                          t                              /                                                              (                                                                  2                                  ⁢                                                                      σ                                    x                                                                                                  )                                                                                                              )                                                2                                                              }                                    /                                                                                                                          ⁢                                                (                                      1                    +                                          (                                              t                        /                                                  τ                          x                                                                    )                                                        )                                ]                                                                        [                  Mathematical          ⁢                      xe2x80x83                    ⁢          Expression          ⁢                      xe2x80x83                    ⁢          33                ]            
Mathematical Expression 34
Gx(t)={B(k, t)+B(xe2x88x92k, t)+4B(0, t)}/{B(k, 0)+B(xe2x88x92k, 0)+4B(0, 0)}xe2x80x83xe2x80x83[Mathematical expression 34]
Because "sgr"x less than  less than V/xcfx84x from mathematical expression 4, approximation of mathematical expression 33 can be performed as in mathematical expression 35 where j is imaginary unit.
Mathematical Expression 35
B(k, t)=xcfx801/2"sgr"xxe2x88x921xc3x97exp[jkVtxe2x88x92k2Dtxe2x88x92{Vt/(2"sgr"x)}2]xe2x80x83xe2x80x83[Mathematical expression 35]
From mathematical expressions 34 and 35, Gx (t) is obtained by mathematical expression 36.
Mathematical Expression 36
Gx(t)=(⅓)exp[xe2x88x92k2Dtxe2x88x92{Vt/(2"sgr"x)}2]xc3x97{cos (2xcfx80Vt/P)+1}+(⅔)exp[{Vt/(2"sgr"x)}2]xe2x80x83xe2x80x83[Mathematical expression 35]
The right member of mathematical expression 36 is the same as the right member of mathematical expression 7. Meanwhile, Gy (t) and Gz (t) are directly obtained from mathematical expressions 26, 28, and 30 and Gi (t) is expressed by mathematical expression 37. Because 1/(k2D)  less than  less than xcfx84i (i=y, z) from mathematical expression 5, Gy (t) and Gz (t) can be regarded as being 1 with regard to t before Gx (t) vanishes. Hence, from mathematical expressions 36 and 37, we can get mathematical expression 7.
Mathematical Expression 37
Gi(t)={(1+(t/xcfx84i)}xe2x88x921/2(i=y, z)xe2x80x83xe2x80x83[Mathematical expression 35]